THE NONDETERMINISTIC COMPUTATION OF FUNCTIONS
Abstract
This thesis is a study of how the use of nondeterministic Turing machines to compute functions provides a framework for the investigation of the inherent time complexity of a large class of problems. The ability to mathematically address the complexity of nondeterministic algorithms for the computation of functions as opposed to the traditional recognition of sets not only enables a more natural and direct formulation of extant problems of interest, but also potentially enlarges the class of problems whose solution in polynomial time would imply P = NP. The notions of polynomial time oracular and transformational reducibilities are extended to reductions between functions, called T-reducibility and K-reducibility respectively, and the resulting relationships are shown to be consistent with those obtained for sets using set reductions. Having defined functional reducibilities it is shown that the Meyer-Stockmeyer polynomial time hierarchy can be extended to a hierarchy of functions, the (SIGMA) classes of the set hierarchy corresponding to what are denoted (LAMDA) classes of the function hierarchy. Sets which are complete with respect to T-reducibility are found in all (LAMDA) levels of the function hierarchy, and functions which are complete with respect to K-reducibility are found in all P((LAMDA)) classes. Finally, a class of problems, called selection problems, are characterized in terms of polynomially bounded operators on solution sets. This class of problems includes exact problems (optimization problems) and unique solution problems. It is shown that there is a tight complexity relationship between the selection problem and its corresponding recognition problem, extending a result of Papadimitriou and Yannakakis. Moreover, it is shown that as an operator on functions, selection can never be weak: an operator defined from any selector is strong enough, when used in conjunction with deterministic polynomial time oracular computation, to climb the extended hierarchy.
Degree
Ph.D.
Subject Area
Computer science
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